Given equation 7.3 of Sutton and Barto's book for convergence of TD(n):
$\max_s|\mathbb{E}_\pi[G_{t:t+n}|S_t = s] - v_\pi(s)| \leqslant \gamma^n \max_s|V_{t+n-1}(s) - v_\pi(s)|$
$\textbf{PROBLEM 1}$ : Why is this error $|\mathbb{E}_\pi[G_{t:t+n}|S_t = s] - v_\pi(s)|$ compared with the error $|V_{t+n-1}(s) - v_\pi(s)|$.
There can be two other logical comparisons for the convergence of algorithm($TD(n)$):
1) If we compare and say that $V_{t+n-1}(s)$ is more close to $v_\pi(s)$ than $V_{t+n-2}(s)$ i.e we compare $|V_{t+n-1}(s) - v_\pi(s)|$ with $|V_{t+n-2}(s) - v_\pi(s)|$
2) We can also compare $|\mathbb{E}_\pi[G_{t:t+n}|S_t = s] - v_\pi(s)|$ with $|\mathbb{E}_\pi[V_{t+n-1}(S_t)|S_t = s] - v_\pi(s)|$ to show that $\mathbb{E}_\pi[G_{t:t+n}|S_t = s]$ is better than $\mathbb{E}_\pi[V_{t+n-1}(S_t)|S_t = s]$ hence moving $V_{t+n-1}(S_t)$ towards $G_{t:t+n}$(as done in eq 7.2) can lead to convergence.
$\textbf{PROBLEM 2}$ : Are the above 2 methods of comparison for testing convergence correct.
Equations for reference:
Eq 7.1: $G_{t:t+n} = R_{t+1} + \gamma R_{t+2} +......+\gamma^{n-1}R_{t+n} + \gamma^{n}V_{t+n-1}(S_{t+n})$
Eq 7.2: $V_{t+n}(S_t) = V_{t+n-1}(S_t) + \alpha [G_{t:t+n} - V_{t+n-1}(S_t)]$
